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# Computing the Fourier transform of 1/(1+x^2)^2 watch

1. I want to compute the Fourier transform of in . To this end, let . Then

Normally one would proceed via integration by parts, but it seems to be a dead end no matter how you go about it. For instance, we can compute:

Does anyone have a suggestion? Certainly, one can compute that , although I doubt this can be of any help.
2. (Original post by RamocitoMorales)
I want to compute the Fourier transform of in .
If you can use standard results, then then use DUTIS on .

I haven't actually tried this myself but I think that'll work.
3. (Original post by atsruser)
If you can use standard results, then then use DUTIS on .

I haven't actually tried this myself but I think that'll work.
If you have the FT for 1/(1+x^2) you should also be able to finish using standard rules for the FT of a derivative and the FT of xf(x).
4. (Original post by atsruser)
If you can use standard results, then then use DUTIS on .

I haven't actually tried this myself but I think that'll work.
What is "DUTIS"?
5. (Original post by RamocitoMorales)
What is "DUTIS"?
Differentiation under the integral sign.
6. (Original post by RamocitoMorales)
I want to compute the Fourier transform of in . To this end, let . Then

As a couple of alternatives:

1. You can try a contour integral approach - not sure how tricky that is in this case though. However in this post:

https://www.thestudentroom.co.uk/sho...&postcount=313

TeeEm computes a similar but simpler integral via contour techniques - you could adapt this slightly to find the integral I suggested earlier, then finish via DUTIS.

2. You can note that since your function is even, the FT reduces to then you can follow the working in this post:

https://www.thestudentroom.co.uk/sho...&postcount=107

where Kummer computes precisely that integral via a Laplace transform approach.
7. (Original post by DFranklin)
If you have the FT for 1/(1+x^2) you should also be able to finish using standard rules for the FT of a derivative and the FT of xf(x).
I can't quite see how you get it using those rules - I must be missing some obvious step.
8. (Original post by atsruser)
I can't quite see how you get it using those rules - I must be missing some obvious step.
Diffing gives you the FT of x/(1+x^2)^2, and then you can relate the FT of x/(1+x^2)^2 to the FT of 1(1+x^2)^2 using the "FT of xf(x)" rule.

(I might be getting it wrong, of course, it's almost 30 years since I did this in earnest...)
9. (Original post by atsruser)
2. You can note that since your function is even, the FT reduces to .
Since the Fourier transform is , shouldn't there be a there somewhere?
10. (Original post by RamocitoMorales)
Since the Fourier transform is , shouldn't there be a there somewhere?
Yes, it should be if is your parameter.

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